Order Reduction and μ-Conservation Law for the Non-Isospectral KdV Type Equation a with Variable-Coefficients

Document Type: Research Articles


Department of Mathematics, Broujerd Branch, Islamic Azad University, Broujerd, Iran.


The goal of this paper is to calculate of order reduction of the KdV type
equation and the non-isospectral KdV type equation using the μ-symmetry
method. Moreover we obtain μ-conservation law of the non-isospectral KdV
type equation using the variational problem method.


[1]H. Airault, Rational solutions of Painleve equation, Studies in Applied
Mathematics , 1979, 61, P.31-53.

[2]T. B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long
waves in nonlinear dispersive systems, Trans. R. Soc. (Lond) ser.A, 1992,
272, P.234-356.

[3]W. Bluman, F. Cheviakov, C. Anco, Construction of conservation law:
how the direct method generalizes Nother's theorem, Group analysis of
differential equations and integrability, 2009, 12 P.1-23.

[4] J.L. Bona, Bryant, P.J., A mathematical model for long waves generated
by wave makers in nonlinear dispersive systems, Proc. Cambridge Phil.
Soc., 1973, 4, P.12-34.

[5]G. Cicogna, G. Gaeta, P. Morando, On the relation between standard and
mu-symmetries for PDEs, J. Phys. A. , 2004, 37, P. 9467-9486.

[6]G. Cicogna, G. Gaeta, Norther theorem for mu-symmetries, J. Phys. A.,
2007, 40, P.11899-11921.

[7] G. Gaeta, P. Morando, On the geometry of lambda-symmetries and PDEs
reduction, J. Phys. A., 2004, 37, P.6955-6975.

[8] G. Gaeta, Lambda and mu-symmetries, SPT2004 , World Scienti c,
Singapore, 2005.

[9] KH. Goodarzi, M.Nadja khah, mu-symmetry and mu-conservation law for
the extended mKdV equation, JNMP (2014),3, P.371-381.

[10]D.J.Korteweg, G. de Vries, On the change of form of long waves advancing
in a rectangular canal and on a new type of long stationary waves, Phil.
Mag. 1895, 39, P.422-443.

[11]A. Kudryashov, I. Sinelshchikov, A note on the Lie symmetry analysis and
exact solutions for the extended mKdV equation, Acta. Appl.Math., 2011,
113, 41-44.

[12]C. Muriel, J.L. Romero, New methods of reduction for ordinary di erential
equation, IMA J. Appl. Math., 2001, 66, P.111-125.

[13]C. Muriel, J.L. Romero, C-symmetries and reduction of equation
without Lie point symmetries, J. Lie Theory, 2003, 13, P.167-188.

[14]C. Muriel, J.L. Romero, P.J. Olver, Variationl C-symmetries and EulerLagrange
equations, J. Di . Eqs, 2006, 222, P.164-184.

[15]C. Muriel, J.L. Romero, Prolongations of vector fields and the invariantsby-derrivation
property, Theor. Math. Phys., 2002, 133,P.1565-1575.

[16]P.J. Olver, Applications of Lie Groups to Differential Equations, New
York, 1986.

[17]C. Vasconcellos, P. N. da Silva, Stabilization of the linear Kawahara
equation, Applied and Computational Mathematics, 2015, 3, P. 45-67.